What is the solution to problem 6 in the Combinatorics seating problem?

In summary, there are 5040 ways to seat 8 people around a circular table, 32 ways to seat 5 couples at a rectangular table, 48 ways to seat 6 people in a row of 6 chairs if 2 people refuse to sit next to each other, 362,880 ways to seat 10 people at a round table if 2 people must sit across from each other, and 24 ways to seat 6 people at a rectangular table if 3 people must sit on each side.
  • #1
EvLer
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Hello,
so this is what I am stuck on:
In how many different ways can you seat 11 men and 8 women in a row so that no two women are to sit next to each other.

I know it's going to be combination and not permutation and that total number of seating them is 11!*8! but that is as far as I could get :frown:
 
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  • #2
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1. How many ways can 8 people be seated around a circular table?

There are (8-1)! = 7! = 5040 ways to seat 8 people around a circular table, since the circular arrangement allows for rotations.

2. In how many ways can 5 couples be seated at a rectangular table if each couple must sit together?

There are 2^5 = 32 ways to seat 5 couples at a rectangular table, since each couple can be arranged in 2 ways (either sitting next to each other or across from each other) and the couples can be arranged in any order.

3. How many ways can 6 people be seated in a row of 6 chairs if 2 people refuse to sit next to each other?

There are 4! x 2! = 48 ways to seat 6 people in a row of 6 chairs if 2 people refuse to sit next to each other. The first person can be seated in any of the 6 chairs. The second person can be seated in any of the remaining 5 chairs. The remaining 4 people can be seated in 4! = 24 ways. However, since the 2 people who refuse to sit next to each other can be seated in either order, we multiply by 2! = 2.

4. How many ways can 10 people be seated at a round table if 2 people must sit across from each other?

There are (10-1)! = 9! = 362,880 ways to seat 10 people at a round table if 2 people must sit across from each other. We treat the 2 people who must sit across from each other as a single unit, so there are 9 seats remaining to be filled by the other 8 people.

5. In how many ways can 6 people be seated at a rectangular table if 3 people must sit on each side?

There are 2 x 3! x (3-1)! = 24 ways to seat 6 people at a rectangular table if 3 people must sit on each side. The first person can sit on any of the 6 chairs. The second and third person must sit on either side of the first person, so there are 2 ways to arrange them. The remaining 3 people can be seated in 3! = 6 ways on the other side of the table. However, since the 3 people on the other side can be arranged in any order, we multiply by (3-1)! = 2.

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